1. Which among 2^(1/2),3^(1/3),4^(1/4),6^(1/6) and 12^(1/12) is the largest?
(a) 2^(1/2)
(b) 3^(1/3)
(c) 4^(1/4)
(d) 6^(1/6)
2. Consider a sequence where the nth term, tn=n/(n+2),n=1,2,……. . The value of t3×t4×t5×……..×t53 equals-
(a) 2/495
(b) 2/477
(c) 12/55
(d) 1/1485
3. If 1/b=1/3,b/c=2,c/d=1/2,d/e=3 and e/f=1/4, then what is the value of abc/def ?
(a) 3/8
(b) 27/8
(c) 3/4
(d) 27/4
4. If x=-0.5, then which of the following has the smallest value?
(a) 2^(1/x)
(b) 1/x
(c) 1/x^2
(d) 2^x
5. The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can possibly be one of these four numbers?
(a) 21
(b) 25
(c) 41
(d) 67
6. When you reverse the digits of the number 13, the number increases by 18. How many other two digit numbers increase by 18 when their digits are reversed?
(a) 5
(b) 6
(c) 7
(d) 8
7. The number of employees in Obelix Menhir Co. is a prime number and is less than 300. The ratio of the number of employees who are graduates and above, to that of employees who are not, can possibly be
(a) 101 : 88
(b) 87 : 100
(c) 110 : 111
(d)97 : 84
8. The rightmost non-zero digit of the number 30^2720 is
(a) 1
(b) 3
(c) 7
(d) 9
9. Let x=√(4+√(4-√(4+√(4-(……..) ∞)) ) ) . Then, x equals
(a) 3
(b) ((√13-1)/2)
(c) ((√13+1)/2)
(d) √13
10. A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is
(a) 10
(b) 12
(c) 14
(d) 16
Answers:
1. (b); LCM of 2, 3, 4, 6, 12=12
(2^1/2)^12, (3^1/3)^12, (4^1/4)^12, (6^1/6)^12, (12^1/12)^12
=(2)^6, (3)^4, (4)^3, (6)^2, (12)^1
=64, 81, 64, 36, 12
Hence, 3^(1/3) is the largest.
2. Given t_n=n/(n+2), n=1,2,….
Therefore, t_3=3/5, t_4=4/6, t_5=5/7, …., t^53=53/55
Therefore t_3 × t_4 × t_5 ×….×t_53
=3/5 × 4/6 × 5/7 × 6/8 ×…..×51/53 × 52/54 × 53/55
=(3×4)/(54×55) = 2/495
3. (a); Given that a/b=1/3, b/c=2, c/d=1/2, d/e=3 and e/f=1/4
Therefore a/b × b/c × c/d = 1/3 × 2 × 1/2= 1/3
a/d=a/3 and c/d × d/e=1/2 × 3 = 3/2
c/e=3/2
and e/f × d/e × b/c × c/d=1/4 × 3 × 2 × 1/2 = 3/4
b/f= 3/4
therefore, abc/ def= a/d × c/e × b/f = 1/3 × 3/2 × ¾ = 3/8
4. (b); Using options, we can solve the question easily.
Put x=- ½
(a) 2^(-2)= ¼
(b) 1/(- ½) = -2
(c) 1/(- a/2)^2 = 4
(d) 2^(-1/2)=1/√2
(e) 1/√-(1/2) = √2
5. (c); Using options, we find that four consecutive odd numbers are 37, 39, 41 and 43. The sum of these 4 numbers is 160, when divided by 10, we get 16 which is a perfect square.
Therefore, 41 is one of the odd numbers.
6. (b); Let the number be (10x+y), so when the digits of number are reversed the number becomes (10y+x). According the question, (10y+x)-(10x+y)=18
9(y-x)=18 y-x=12
So, the possible pairs of (x, y) are
(1, 3), (2, 4), (3, 5), (4, 6), (5, 7), (6, 8) and (7, 9).
But we want the number other than 13. Thus, there are 6 possible numbers ie, 24, 35, 46, 57, 68, 79.
So, total number of possible numbers are 6.
7. (d); Using options, we find that sum of numerator and denominator of 97 : 84 is (97+84)=181 which is a prime number. Hence, it is the appropriate answer.
8. (a) [(30^4]^680 hence, the rightmost non-zero digit is 1.
9. (c); x=(√(4+√(4-x))
(x^2-4)=4+√(4-x)
Now, put the values from option only option (c) satisfies the condition.
10. (b); Let the rectangle has m and n tiles along its length and breadth respectively.
The number of white tiles
W=2m+2(n-2)=2(m+n-2)
And the number of red tiles=R=mn-2(m+n-2)
Given W=R 4(m+n-2)=mn
mn-4m-4n=-8
(m-4) (n-4)=8
m-4=8 or 4
M=12 or 8
Therefore 12 suits the option.
1. (b); LCM of 2, 3, 4, 6, 12=12
(2^1/2)^12, (3^1/3)^12, (4^1/4)^12, (6^1/6)^12, (12^1/12)^12
=(2)^6, (3)^4, (4)^3, (6)^2, (12)^1
=64, 81, 64, 36, 12
Hence, 3^(1/3) is the largest.
2. Given t_n=n/(n+2), n=1,2,….
Therefore, t_3=3/5, t_4=4/6, t_5=5/7, …., t^53=53/55
Therefore t_3 × t_4 × t_5 ×….×t_53
=3/5 × 4/6 × 5/7 × 6/8 ×…..×51/53 × 52/54 × 53/55
=(3×4)/(54×55) = 2/495
3. (a); Given that a/b=1/3, b/c=2, c/d=1/2, d/e=3 and e/f=1/4
Therefore a/b × b/c × c/d = 1/3 × 2 × 1/2= 1/3
a/d=a/3 and c/d × d/e=1/2 × 3 = 3/2
c/e=3/2
and e/f × d/e × b/c × c/d=1/4 × 3 × 2 × 1/2 = 3/4
b/f= 3/4
therefore, abc/ def= a/d × c/e × b/f = 1/3 × 3/2 × ¾ = 3/8
4. (b); Using options, we can solve the question easily.
Put x=- ½
(a) 2^(-2)= ¼
(b) 1/(- ½) = -2
(c) 1/(- a/2)^2 = 4
(d) 2^(-1/2)=1/√2
(e) 1/√-(1/2) = √2
5. (c); Using options, we find that four consecutive odd numbers are 37, 39, 41 and 43. The sum of these 4 numbers is 160, when divided by 10, we get 16 which is a perfect square.
Therefore, 41 is one of the odd numbers.
6. (b); Let the number be (10x+y), so when the digits of number are reversed the number becomes (10y+x). According the question, (10y+x)-(10x+y)=18
9(y-x)=18 y-x=12
So, the possible pairs of (x, y) are
(1, 3), (2, 4), (3, 5), (4, 6), (5, 7), (6, 8) and (7, 9).
But we want the number other than 13. Thus, there are 6 possible numbers ie, 24, 35, 46, 57, 68, 79.
So, total number of possible numbers are 6.
7. (d); Using options, we find that sum of numerator and denominator of 97 : 84 is (97+84)=181 which is a prime number. Hence, it is the appropriate answer.
8. (a) [(30^4]^680 hence, the rightmost non-zero digit is 1.
9. (c); x=(√(4+√(4-x))
(x^2-4)=4+√(4-x)
Now, put the values from option only option (c) satisfies the condition.
10. (b); Let the rectangle has m and n tiles along its length and breadth respectively.
The number of white tiles
W=2m+2(n-2)=2(m+n-2)
And the number of red tiles=R=mn-2(m+n-2)
Given W=R 4(m+n-2)=mn
mn-4m-4n=-8
(m-4) (n-4)=8
m-4=8 or 4
M=12 or 8
Therefore 12 suits the option.
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